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obviously is true. The upshot is that the replacement of nun*.
bers by their corresponding classes does what is required wlien
we add the definition of 'multiplication' : (m) x (n) is defined to
be the class (mn); (2) x (6) = (12). Notice that the last is a
definition; it is not meant to follow from the meanings of ($1
and (n).

Dedekind's ideals for algebraic numbers are a generalizaticm
of what precedes. Following his usual custom Dedekind gave
an abstract definition, that is, a definition based upon essential
properties rather than one contingent upon some particular
mode of representing, or picturing, the thing defined.

Consider the set (or class) of all algebraic integers in a given
algebraic number field. In this all-inclusive set will be subsets.
A subset is called an ideal if it has the two following properties.

A.  The mm and difference of any two integers in the subset
are also in the subset.

B.  If any integer in the subset be multiplied by any intega
in the all-inclusive set, the resulting integer is in the subset.

An ideal is thus an infinite class of integers. It will be seen
readily that (m), (n), . . . , previously defined, are ideals accord
ing to A, B. As before, if one ideal contains another, the fiisi
is said to divide the second.

It can be proved that every ideal is a class of integers all d
which are of the form

where al9 a2, ... , an axe fixed integers of the field of degree 
concerned, and each of xit 02 2, ... , xn may be any integer what-
ever in the field. This being so, it is convenient to symbolize as
ideal by exhibiting only the fixed integers, a19 aa . . . , an, an<
this is done by writing (ax> aa, . . .  an) as the symbol of tfc
ideal. The order in which ai9 a^ , .  , an are written in the
symbol is immaterial*
'Multiplication* of ideals must now be defined: the product <S
the two ideals (<zls . . . , an), (bi9 ... , &n) is the ideal whoa
symbol is (a^, ... A   -  fl&n) ^ wnicn ^ possible1
products, flj&u etc., obtained by multiplying an integer in the!
first symbol by an integer in the second occur. For example, the